Bulletin of the
Korean Mathematical Society BKMS

Let $r \geq 2$, and let $ k_i \geq 2$ for $1 \leq i \leq r$. Mixed van der Waerden's theorem states that there exists a least positive integer $w= w(k_1, k_2, k_3, \dots, k_r;r)$ such that for any $n \geq w$, every $r$-colouring of $[1,n]$ admits a $k_i$-term arithmetic progression with colour $i$ for some $i \in [1,r]$. For $k \geq 3$ and $r \geq 2$, the mixed van der Waerden number $w(k,2,2, \dots, 2;r)$ is denoted by $w_2(k;r)$. B. Landman and A. Robertson \cite{vdw5} showed that for $k < r < \frac{3}{2} (k-1)$ and $r \geq 2k+2$, the inequality $w_2(k;r) \leq r(k-1)$ holds. In this note, we establish some results on $w_2(k;r)$ for $2 \leq r \leq k$.

Keywords: Mixed van der Waerden number, Ramsey theory on the integers

MSC numbers: Primary 05C57, 05C80

Supported by: http://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.b200718